Question

Prove that 1-cosA/1+cosA=cosecA-cotA

Answer 1

$\mathfrak{\underline{\underline{\green{ Solution:-}}}}$

$\underline{\pink{To \: prove}}$

$\mathbf{\sqrt{\dfrac{1-cosA}{1+cosA}}= cosecA - cotA}$

$\underline{\pink{ proof}}$

Let us consider LHS,

LHS :

$\mathbf{ = \sqrt{\dfrac{1-cosA}{1+cosA}}}$

By rationalising the denominator

$\mathbf{ = \sqrt{\dfrac{(1-cosA)\:(1-cosA)}{(1+cosA)(1-cosA)}}}$

$\mathbf{ = \sqrt{\dfrac{{(1-cosA)}^{2}}{1-{cos}^{2}A}}}$

$\mathbf{ = \sqrt{\dfrac{{(1-cosA)}^{2}}{{sin}^{2}A}}}$

On cancelling square and square root,

$\mathbf{ =\dfrac{1-cosA}{1+cosA}}$

$\mathbf{ = \dfrac{1}{sinA} -\dfrac{cosA}{sinA}}$

$\boxed{\red{\dfrac{1}{sinA} = cosecA}}$

$\boxed{\red{\dfrac{cosA}{sinA} = cotA}}$

Therefore,

$\mathbf{ = CosecA-cot A}$

$\mathbf{RHS = CosecA-cot A}$

Here,

$\boxed{\</strong><strong>pink</strong><strong>{</strong><strong>L</strong><strong>H</strong><strong>S</strong><strong> </strong><strong>=</strong><strong> </strong><strong>RHS</strong><strong>} }$

Hence proved!